Ολοκληρώματα
Ολοκληρώματα integrals Ακολουθεί κατάλογος ολοκληρωμάτων. Ολοκληρώματα Integration is one of the two basic operations in calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives. We use C'' for an "Αυθαίρετη Ολοκληρωτική Σταθερά" (arbitrary constant of integration) that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives. Απλές Συναρτήσεις Άρρητες Συναρτήσεις : \int {dx \over \sqrt{a^2-x^2}} = \sin^{-1} {x \over a} + C : \int {-dx \over \sqrt{a^2-x^2}} = \cos^{-1} {x \over a} + C : \int {dx \over x \sqrt{x^2-a^2}} = {1 \over a} \sec^{-1} + C : \int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C : \int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C : \int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C : \int \sec^2 x \, dx = \tan x + C : \int \csc^2 x \, dx = -\cot x + C : \int \sec{x} \, \tan{x} \, dx = \sec{x} + C : \int \csc{x} \, \cot{x} \, dx = -\csc{x} + C : \int \sin^2 x \, dx = \frac{1}{2}(x - \frac{\sin 2x}{2} ) + C = \frac{1}{2}(x - \sin x\cos x ) + C : \int \cos^2 x \, dx = \frac{1}{2}(x + \frac{\sin 2x}{2}) + C = \frac{1}{2}(x + \sin x\cos x ) + C : \int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C :: (see integral of secant cubed) : \int \sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int \sin^{n-2}{x} \, dx : \int \cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int \cos^{n-2}{x} \, dx : \int \arctan{x} \, dx = x \, \arctan{x} - \frac{1}{2} \ln{\left| 1 + x^2\right|} + C Υπερβολικές Συναρτήσεις : \int \sinh x \, dx = \cosh x + C : \int \cosh x \, dx = \sinh x + C : \int \tanh x \, dx = \ln| \cosh x | + C : \int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C : \int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C : \int \coth x \, dx = \ln| \sinh x | + C : \int \mbox{sech}^2 x\, dx = \tanh x + C Αντίστροφες Υπερβολικές Συναρτήσεις : \int \operatorname{arcsinh}\, x \, dx = x\, \operatorname{arcsinh}\, x - \sqrt{x^2+1} + C : \int \operatorname{arccosh}\, x \, dx = x\, \operatorname{arccosh}\, x - \sqrt{x^2-1} + C : \int \operatorname{arctanh}\, x \, dx = x\, \operatorname{arctanh}\, x + \frac{1}{2}\log{(1-x^2)} + C : \int \operatorname{arccsch}\,x \, dx = x\, \operatorname{arccsch}\, x+ \log{\left+ 1\right)\right} + C : \int \operatorname{arcsech}\,x \, dx = x\, \operatorname{arcsech}\, x- \arctan{\left(\frac{x}{x-1}\sqrt{\frac{1-x}{1+x}}\right)} + C : \int \operatorname{arccoth}\,x \, dx = x\, \operatorname{arccoth}\, x+ \frac{1}{2}\log{(x^2-1)} + C Definite integrals lacking closed-form antiderivatives There are some functions whose antiderivatives ''cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below. : \int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi (see also Gamma function) : \int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi (the Gaussian integral) : \int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6} (see also Bernoulli number) : \int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15} : \int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2} : \int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2} (if n'' is an even integer and \scriptstyle{n \ge 2} ) : \int_0^\frac{\pi}{2}\sin^n{x}\,dx=\int_0^\frac{\pi}{2}\cos^n{x}\,dx=\frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n} (if \scriptstyle{n} is an odd integer and \scriptstyle{n \ge 3} ) : \int_0^\infty\frac{\sin^2{x}}{x^2}\,dx=\frac{\pi}{2} : \int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z) (where \Gamma(z) is the Gamma function) : \int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx=\sqrt{\frac{\pi}{a}}\exp\left\frac{b^2-4ac}{4a}\right (where \expu is the exponential function e^u , and a>0 ) : \int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x) (where I_{0}(x) is the modified Bessel function of the first kind) : \int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right) : \int_{-\infty}^{\infty}{(1 + x^2/\nu)^{-(\nu + 1)/2}dx} = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2))}\, , \nu > 0\, , this is related to the probability density function of the Student's t-distribution) The method of exhaustion provides a formula for the general case when no antiderivative exists: : \int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} ). Historical development of integrals A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meyer Hirsch in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David de Bierens de Haan. A new edition was published in 1862. These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by de Bierens are denoted by BI. Since 1968 there is the Risch algorithm for determining indefinite integrals. Other lists of integrals Gradshteyn and Ryzhik contains a large collection of results. Other useful resources include the ''CRC Standard Mathematical Tables and Formulae and Abramowitz and Stegun. A&S contains many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. There are several web sites which have tables of integrals and integrals on demand. Πίνακες Ολοκληρωμάτων * S.O.S. Mathematics: Tables and Formulas * Paul's Online Math Notes Εσωτερική Αρθρογραφία *ολοκλήρωμα * Βιβλιογραφία * Besavilla: Engineering Review Center, Engineering Mathematics (Formulas), Mini Booklet * Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. * I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.) * Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. (Many earlier editions as well.) * Meyer Hirsch, Integraltafeln, oder, Sammlung von Integralformeln (Duncker un Humblot, Berlin, 1810) * Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln] * David de Bierens de Haan, Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862) * Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899) Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] *